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Theorem opnnei 23112
Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)
Assertion
Ref Expression
opnnei (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei
StepHypRef Expression
1 0opn 22894 . . . . 5 (𝐽 ∈ Top → ∅ ∈ 𝐽)
21adantr 479 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3 eleq1 2814 . . . . 5 (𝑆 = ∅ → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
43adantl 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
52, 4mpbird 256 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆𝐽)
6 rzal 4503 . . . 4 (𝑆 = ∅ → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
76adantl 480 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
85, 72thd 264 . 2 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9 opnneip 23111 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝐽𝑥𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1093expia 1118 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑥𝑆𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1110ralrimiv 3135 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝐽) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1211ex 411 . . . 4 (𝐽 ∈ Top → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1312adantr 479 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14 df-ne 2931 . . . . . 6 (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15 r19.2z 4489 . . . . . . 7 ((𝑆 ≠ ∅ ∧ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1615ex 411 . . . . . 6 (𝑆 ≠ ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1714, 16sylbir 234 . . . . 5 𝑆 = ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18 eqid 2726 . . . . . . . 8 𝐽 = 𝐽
1918neii1 23098 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 𝐽)
2019ex 411 . . . . . 6 (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2120rexlimdvw 3150 . . . . 5 (𝐽 ∈ Top → (∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2217, 21sylan9r 507 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2318ntrss2 23049 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
2423adantr 479 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25 vex 3466 . . . . . . . . . . . . 13 𝑥 ∈ V
2625snss 4784 . . . . . . . . . . . 12 (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
2726ralbii 3083 . . . . . . . . . . 11 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28 dfss3 3967 . . . . . . . . . . . . 13 (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
2928biimpri 227 . . . . . . . . . . . 12 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3029adantl 480 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3127, 30sylan2br 593 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3224, 31eqssd 3996 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
3332ex 411 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
3425snss 4784 . . . . . . . . . . . 12 (𝑥𝑆 ↔ {𝑥} ⊆ 𝑆)
35 sstr2 3985 . . . . . . . . . . . . . 14 ({𝑥} ⊆ 𝑆 → (𝑆 𝐽 → {𝑥} ⊆ 𝐽))
3635com12 32 . . . . . . . . . . . . 13 (𝑆 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3736adantl 480 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3834, 37biimtrid 241 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑥𝑆 → {𝑥} ⊆ 𝐽))
3938imp 405 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → {𝑥} ⊆ 𝐽)
4018neiint 23096 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝐽𝑆 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41403com23 1123 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽 ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42413expa 1115 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4339, 42syldan 589 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4443ralbidva 3166 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4518isopn3 23058 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑆𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
4633, 44, 453imtr4d 293 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
4746ex 411 . . . . . 6 (𝐽 ∈ Top → (𝑆 𝐽 → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽)))
4847com23 86 . . . . 5 (𝐽 ∈ Top → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
4948adantr 479 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
5022, 49mpdd 43 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
5113, 50impbid 211 . 2 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
528, 51pm2.61dan 811 1 (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  wss 3946  c0 4322  {csn 4623   cuni 4905  cfv 6546  Topctop 22883  intcnt 23009  neicnei 23089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-top 22884  df-ntr 23012  df-nei 23090
This theorem is referenced by:  neiptopreu  23125  flimcf  23974
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